which depends on both the density \(k_{\mathrm {F}}\)  and the momentum \(|\mathbf {k}|\) .
The symmetry potential and the second-order potential could be obtained straightforwardly by expanding the above expression around \(\delta =0\) . Putting them into the relation (REF ) with \(d=3\) , both sides readily give the result \(D^{[4]}k_{\mathrm {F}}^7/27\pi ^2\) , i.e., \(U_{\mathrm {sym,2}}(\rho ,k_{\mathrm {F}})=D^{[4]}k_{\mathrm {F}}^7/27\pi ^2\) .
Other momentum- and density-dependent terms of the Skyrme-like potential of Ref. [1]} could be checked similarly. Another potential application of (REF ) is to construct consistent single-nucleon potentials.
For instance, consider a simple nucleon potential depending on the density alone as \(U_J(\rho )=[\alpha +\alpha _{\mathrm {IS}}f( \Gamma _{\vartheta }\tau _3^J\delta )]\widetilde{\rho }^{\vartheta }\)  where \(\widetilde{\rho }=\rho /\rho _0\)  is the reduced density and \(\Gamma _{\vartheta }\)  is a factor needed to fulfill the identity (REF ).
When using the relation (REF ) the isospin-independent term \(\alpha \)  plays no role and only the isospin-dependent term represented by the \(\alpha _{\mathrm {IS}}\)  is relevant.
Consequently, we have \(U_{\mathrm {sym}}(\rho )=\alpha _{\mathrm {IS}}\Gamma _{\vartheta }f^{\prime }(0)\widetilde{\rho }^{\vartheta }\)  where the derivative of \(f\)  is taken with respect to the isospin asymmetry \(\delta \) , and similarly \(U_{\mathrm {sym},2}(\rho )=2^{-1}\alpha _{\mathrm {IS}}\Gamma _{\vartheta }^2f^{\prime \prime }(0)\widetilde{\rho }^{\vartheta }\) .
Putting the \(U_{\mathrm {sym}}(\rho )\)  and \(U_{\mathrm {sym,2}}(\rho )\)  into (REF ) gives the relation,
\(\Gamma _{\vartheta }=\frac{f^{\prime }(0)}{f^{\prime \prime }(0)}\cdot ({\vartheta }-1).\) 
